Police said the green Fiat Punto was in a collison with a BMW which had been driving alongside it before changing lanes.
The Punto went off the road, through a fence and into the construction site, hitting a lamppost on the way.
The driver of the BMW failed to stop at the scene while the driver of the Punto was taken to hospital with minor injuries and shock.
The Northampton Chronicle & Echo said the site was for the University of Northampton’s Innovation Cube building where Kier is the main contractor on the £7.3m job.